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Math Help - Limits / Series question 2

  1. #1
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    Limits / Series question 2

    Let a1 = 1 and an = n(an-1 + 1 ) for n=2,3,.....

    Define

    P = (1+ 1/a1 )(1+1/a2)......(1+1/an )

    The Limit P when n tends to infinity is?
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  2. #2
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    Quote Originally Posted by champrock View Post
    Let a1 = 1 and an = n(an-1 + 1 ) for n=2,3,.....

    Define

    P = (1+ 1/a1 )(1+1/a2)......(1+1/an )

    The Limit P when n tends to infinity is?
    Step 1. Work out the first few terms to get an idea of what is happening. You should find that the first few values of a_n are 1, 4, 15, 64, 325, 1956, and that the corresponding value of P is \tfrac21.\tfrac54.\tfrac{16}{15}.\tfrac{65}{64}.\t  frac{326}{325} .\tfrac{1957}{1956}. This simplifies to \frac{1957}{6!} \approx2.718, which looks suspiciously close to e.

    Now see if you can turn those observations into proofs. You need to show first that the product (1+ 1/a_1 )(1+1/a_2)\ldots(1+1/a_n ) is equal to \frac{a_n+1}{n!}. Then you want to show that \lim_{n\to\infty}\frac{a_n}{n!} = e.

    Hint for that last part: let b_n = \frac{a_n}{n!}. Show that b_n = b_{n-1}+\frac1{(n-1)!}, and deduce that b_n = 1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\ldots+\frac1{(  n-1)!}.
    Last edited by Opalg; April 3rd 2009 at 07:13 AM. Reason: corrected error
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  3. #3
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    answer is given as (e+1). ..
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